From algebraic curve to minimal surface and back

TL;DR

This paper derives the Lax operator for a broad class of classical minimal surface solutions in AdS3, linking algebraic curves to hyperelliptic surfaces associated with Wilson loops in N=4 SYM theory.

Contribution

It introduces a method to derive the Lax operator for minimal surfaces in AdS3 and confirms the algebraic curve matches the hyperelliptic surface of the solutions.

Findings

Lax operator derived for minimal surface solutions

Algebraic curve identified as hyperelliptic surface

Verification of the curve-surface correspondence

Abstract

We derive the Lax operator for a very large family of classical minimal surface solutions in $AdS_3$ describing Wilson loops in $\mathcal{N}=4$ SYM theory. These solutions, constructed by Ishizeki, Kruczenski and Ziama, are associated with a hyperellictic surface of odd genus. We verify that the algebraic curve derived from the Lax operator is indeed none-other than this hyperelliptic surface.